
By the results contained in [1]}, when \(q\)  is large enough, there always exists a \(\sigma \) -linearized polynomial with maximum kernel of form
\( -x^{\sigma ^d}+a_{d-2}x^{\sigma ^{d-2}}+\ldots +a_0x\in \mathcal {L}_{n,q,\sigma }. \) 
Thus, there could be room for \(\sigma \) -linearized polynomials with maximum kernel and having some coefficients of high \(\sigma \) -degree equal to zero.

When \(n=d^2-1\) , there are others \(\sigma \) -linearized trinomials with maximum kernel different from those found in Theorem REF , as suggested by Proposition REF . Determine the remaining family of \(\sigma \) -linearized polynomials with maximum kernel when \(n=d^2-1\) .

More generally, is it possible to manipulate the equations in Theorem REF  to get some more examples of linearized trinomials with maximum kernel also when \(g \in \lbrace 2,\ldots ,d-2\rbrace \)  and \(d\ge 4\) ?

